When you animate the depth Scale of a Set it will scale all of the geometry of an object. This will result in an unflattering flattening (!) of the bevel if the depth scale is reduced too far.

Here is a little trick you can do when you absolutely positively have to animate the depth and want the bevel to stay rock solid.

In a nutshell you make two objects. One contains only the bevel. The other contains only edge thickness with no bevel. By positioning these two objects in front of each other, and putting each in a different Set, you can animate the depth Scale of the edge object all the way to zero and the bevel object won't be affected at all.

The secret is, how do you make these two objects?

You open the same Illustrator file twice, then give the bevel object the Straight Flare edge profile. (Edges/ Special Purpose/ Flare, Straight) The depth of the bevel is determined by the depth of this first object. Somewhere between 2 and 10 is all the more depth you'll probably want to give to the bevel. By adjusting the Edge Scale setting you can control the angle of the bevel.

Once you have the bevel done, give the second object the Outline edge profile. (Edges/ Inset Face/ Outline) This profile creates an offset to the front faces. Adjust the depth to your liking, then use the Edge Scale slider to adjust the offset so that the edges of this second object just touch the corners of the front object. Move the second object backward so that it looks like these two objects are one continuous model.
(NOTE: If you have the Pro version, you can use the Basic, Flat Sides edge profile and adjust the offset with the Edge Offset slider.)

Now you can animate the depth Scale of the back object. Note that you may also have to animate the z position of the back object so the front and back objects stay touching.